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\section{Introduction}
Let $\Omega=(a_x, b_x)\times(a_y, b_y) \times (a_y, b_y) \in
\mathbb{R}^3$ with its boundary $\Gamma=\partial \Omega$ and
$J=(0,T]$ be the time interval, $T>0$. The following Initial Boundary
Value Problem (IBVP) is considered:
\begin{eqnarray}
u_t= \nabla\cdot(a\nabla u) + c_g\nabla\cdot(u \nabla g) +  \nonumber \\
a_h\nabla\cdot(u \nabla h) +c(t, x, y) u + f&, \quad t\in J, &\quad 
(x,y,z)\in \Omega \label{eq:heat}\\
u(0,x,y,z) = u_0(x,y,z)\\
u_x=u_y=y_z = 0,&  &\quad (x,y,z) \in \Gamma \\ 
g_x=g_y=g_z = 0, & &\quad (x,y,z) \in\Gamma \\
h_x=h_y=h_z = 0, & &\quad (x,y,z) \in\Gamma 
\end{eqnarray}


\section{ADI Method implementation.}
Alternating Ditection Implicit method is a computationally efficient
scheme, that satisfies the
  following properties:
  \begin{itemize}
\item has accurancy $O(k^2 + h^2)$, $k$ is the time step and $h$ the
spatial step;
    \item is unconditionally stable;
\item number of operations per time step is proportional to the number
of unknowns, $O(M)$
     where $M$ is the number of unknowns.       
  \end{itemize}

\subsection{Douglas-Gunn method}
Let us define the operators:
\begin{eqnarray}
A_1u = -(au_x)_x  - c_g(u g_x)_x - c_h(u h_x)_x  - \frac{1}{3}cu, \nonumber \\
A_2u = -(au_y)_y  - c_g(u g_y)_y - c_h(u h_y)_y  - \frac{1}{3}cu, \nonumber \\
A_3u = -(au_z)_z  - c_g(u g_z)_z - c_h(u h_z)_z  - \frac{1}{3}cu, \nonumber
\end{eqnarray}
then equation %(\ref{eq:heat})
 can be rewitten as:
\begin{eqnarray} 
u_t + A_1u + A_2u  + A_3 u= f. \label{eq:heatA}
\end{eqnarray}
Replacing the operators $A_1$, $A_2$ and $A_3$ (\ref{eq:heatA}) respectively
by their spatial approximations $A_{1h}$, $A_{2h}$ and $A_{3h}$, the proposed Douglas-Gunn method
is:
\begin{align*}
(I+\frac{k}{2}A_{1h})u^* &= (I-\frac{k}{2}A_{1h} - k A_{2h} - k A_{3h})u^n +
k f^{n}, \quad &\text{(x-sweep)}\\
(I+\frac{k}{2}A_{2h})u^{**} &= u^* + \frac{k}{2}A_{2h} u^n , \quad &\text{(y-sweep)}\\
(I + \frac{k}{2} A_{3h})u^{n+1} &= u^{**} + \frac{k}{2}A_{3h} u^n + \frac{k}{2}
(f^{n+1} - f^{n}), \quad &\text{(z-sweep)}.
\end{align*} 

 

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